# Properties

 Label 64.8.a.f Level $64$ Weight $8$ Character orbit 64.a Self dual yes Analytic conductor $19.993$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(1,64)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$19.9926416310$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 44 q^{3} - 430 q^{5} + 1224 q^{7} - 251 q^{9}+O(q^{10})$$ q + 44 * q^3 - 430 * q^5 + 1224 * q^7 - 251 * q^9 $$q + 44 q^{3} - 430 q^{5} + 1224 q^{7} - 251 q^{9} - 3164 q^{11} - 6118 q^{13} - 18920 q^{15} - 16270 q^{17} - 5476 q^{19} + 53856 q^{21} - 1576 q^{23} + 106775 q^{25} - 107272 q^{27} - 122838 q^{29} - 251360 q^{31} - 139216 q^{33} - 526320 q^{35} + 52338 q^{37} - 269192 q^{39} - 319398 q^{41} + 710788 q^{43} + 107930 q^{45} - 284112 q^{47} + 674633 q^{49} - 715880 q^{51} - 296062 q^{53} + 1360520 q^{55} - 240944 q^{57} - 897548 q^{59} + 884810 q^{61} - 307224 q^{63} + 2630740 q^{65} + 4659692 q^{67} - 69344 q^{69} + 2710792 q^{71} - 5670854 q^{73} + 4698100 q^{75} - 3872736 q^{77} + 5124176 q^{79} - 4171031 q^{81} - 1563556 q^{83} + 6996100 q^{85} - 5404872 q^{87} + 11605674 q^{89} - 7488432 q^{91} - 11059840 q^{93} + 2354680 q^{95} + 10931618 q^{97} + 794164 q^{99}+O(q^{100})$$ q + 44 * q^3 - 430 * q^5 + 1224 * q^7 - 251 * q^9 - 3164 * q^11 - 6118 * q^13 - 18920 * q^15 - 16270 * q^17 - 5476 * q^19 + 53856 * q^21 - 1576 * q^23 + 106775 * q^25 - 107272 * q^27 - 122838 * q^29 - 251360 * q^31 - 139216 * q^33 - 526320 * q^35 + 52338 * q^37 - 269192 * q^39 - 319398 * q^41 + 710788 * q^43 + 107930 * q^45 - 284112 * q^47 + 674633 * q^49 - 715880 * q^51 - 296062 * q^53 + 1360520 * q^55 - 240944 * q^57 - 897548 * q^59 + 884810 * q^61 - 307224 * q^63 + 2630740 * q^65 + 4659692 * q^67 - 69344 * q^69 + 2710792 * q^71 - 5670854 * q^73 + 4698100 * q^75 - 3872736 * q^77 + 5124176 * q^79 - 4171031 * q^81 - 1563556 * q^83 + 6996100 * q^85 - 5404872 * q^87 + 11605674 * q^89 - 7488432 * q^91 - 11059840 * q^93 + 2354680 * q^95 + 10931618 * q^97 + 794164 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 44.0000 0 −430.000 0 1224.00 0 −251.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.a.f 1
3.b odd 2 1 576.8.a.z 1
4.b odd 2 1 64.8.a.b 1
8.b even 2 1 16.8.a.a 1
8.d odd 2 1 8.8.a.b 1
12.b even 2 1 576.8.a.y 1
16.e even 4 2 256.8.b.a 2
16.f odd 4 2 256.8.b.g 2
24.f even 2 1 72.8.a.a 1
24.h odd 2 1 144.8.a.a 1
40.e odd 2 1 200.8.a.b 1
40.f even 2 1 400.8.a.p 1
40.i odd 4 2 400.8.c.f 2
40.k even 4 2 200.8.c.c 2
56.e even 2 1 392.8.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.b 1 8.d odd 2 1
16.8.a.a 1 8.b even 2 1
64.8.a.b 1 4.b odd 2 1
64.8.a.f 1 1.a even 1 1 trivial
72.8.a.a 1 24.f even 2 1
144.8.a.a 1 24.h odd 2 1
200.8.a.b 1 40.e odd 2 1
200.8.c.c 2 40.k even 4 2
256.8.b.a 2 16.e even 4 2
256.8.b.g 2 16.f odd 4 2
392.8.a.b 1 56.e even 2 1
400.8.a.p 1 40.f even 2 1
400.8.c.f 2 40.i odd 4 2
576.8.a.y 1 12.b even 2 1
576.8.a.z 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 44$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 44$$
$5$ $$T + 430$$
$7$ $$T - 1224$$
$11$ $$T + 3164$$
$13$ $$T + 6118$$
$17$ $$T + 16270$$
$19$ $$T + 5476$$
$23$ $$T + 1576$$
$29$ $$T + 122838$$
$31$ $$T + 251360$$
$37$ $$T - 52338$$
$41$ $$T + 319398$$
$43$ $$T - 710788$$
$47$ $$T + 284112$$
$53$ $$T + 296062$$
$59$ $$T + 897548$$
$61$ $$T - 884810$$
$67$ $$T - 4659692$$
$71$ $$T - 2710792$$
$73$ $$T + 5670854$$
$79$ $$T - 5124176$$
$83$ $$T + 1563556$$
$89$ $$T - 11605674$$
$97$ $$T - 10931618$$